Higher-Order Linear ODEs

Second-order equations model vibrations and RLC circuits. Learn homogeneous constant-coefficient equations and the auxiliary (characteristic) equation, real, repeated and complex roots, damping regimes, non-homogeneous equations with complementary function + particular integral, undetermined coefficients, variation of parameters, and resonance.

A car's suspension, a building in an earthquake, an RLC filter and a guitar string all obey m y″ + c y′ + k y = F(t). The characteristic roots tell you immediately whether the system oscillates, how fast it decays and whether it can resonate.

An analogy

The characteristic equation is the system's 'personality test': real roots mean sluggish, complex roots mean bouncy, and zero damping means it rings forever.

Homogeneous equations

Try y = e^(rt) → characteristic equation. Two real roots: overdamped; repeated root: critically damped (fastest without overshoot); complex roots: underdamped oscillation at ω_d = ω_n√(1 − ζ²).

Undetermined coefficients

Guess y_p of the same form as the forcing (polynomial, exponential, sin/cos). If the guess duplicates a term in y_c, multiply by t (resonance case).

Variation of parameters

Works for any forcing g(t) (e.g. tan t, 1/t) using the Wronskian of the homogeneous solutions.

Resonance and design

Forcing near ω_n with low damping produces large amplitudes (Tacoma Narrows, machinery vibration). Engineers add damping, detune ω_n or isolate. Higher-order equations follow the same root-based method.

Formulas

• Auxiliary equation: a y″ + b y′ + c y = 0 → a r² + b r + c = 0
• Solutions: Distinct: C₁e^(r₁t) + C₂e^(r₂t); repeated: (C₁ + C₂t)e^(rt); complex α ± jβ: e^(αt)(C₁cos βt + C₂ sin βt)
• Mass–spring–damper: m x″ + c x′ + k x = F(t); ω_n = √(k/m), ζ = c/(2√(km))
• General solution: y = y_c + y_p
• Variation of parameters: y_p = −y₁∫ y₂g/W dt + y₂∫ y₁g/W dt, W = y₁y₂′ − y₂y₁′

Worked example: Underdamped vibration

Given: 2x″ + 8x′ + 50x = 0

Find: Roots, ω_n, ζ, ω_d.

• 2r² + 8r + 50 = 0 → r = −2 ± 4.583j
• ω_n = √25 = 5 rad/s, ζ = 8/(2√100) = 0.4
• ω_d = 4.58 rad/s

Answer: x = e^(−2t)(C₁cos 4.58t + C₂ sin 4.58t)

Worked example: Undetermined coefficients

Given: y″ + 4y = 8 cos t

Find: y_p.

• Try y_p = A cos t
• −A + 4A = 8 → A = 8/3

Answer: y = C₁cos 2t + C₂sin 2t + (8/3) cos t

Common mistakes

• Wrong: Using the same guess when forcing matches y_c. — Right: Multiply by t (or t²).
• Wrong: Applying ICs to y_c alone. — Right: Apply them to y = y_c + y_p.

Applications

• Vehicle suspension: Tuning ζ ≈ 0.2–0.4 for comfort.
• RLC circuits: L q″ + R q′ + q/C = V(t).
• Structures: Earthquake and wind response.

Key takeaways

• Characteristic equation from e^(rt).
• Real/repeated/complex roots → over/critical/under-damped.
• y = y_c + y_p.
• Resonance when forcing matches ω_n.

Going further

• Cauchy–Euler equations: x²y″ + axy′ + by = 0.
• Frequency response: Amplitude ratio vs forcing frequency.
• Multi-DOF systems: Matrix eigenvalue problems (M6).